Regularity of languages defined by formal series with isolated cut point

A formal power series \u3c6 with a real cut point \u3bb defines the language L_{\u3c6,\u3bb} = {\u3c9 08 \u3a3 17 | |\u3c6(\u3c9) > \u3bb}; \u3bb is called isolated if inf_\u3c9 |\u3c6(\u3c9) 12 \u3bb| > 0. In this paper we give conditions for guaranteeing the regularity of L_{\u3c6,\u3bb}, with \u3bb isolated, for two classes of formal series: rational series and Hadamard quotients of rational series. Finally, we provide an explicit representation of the behavior of a subclass of two-way weighted automata in terms of Hadamard quotient of rational series